4-4 Johannes Kepler proposed elliptical paths for the planets about the Sun

The task that Johannes Kepler took on at the beginning of the seventeenth century was to find a model of planetary motion that agreed completely with Tycho’s extensive and very accurate observations of planetary positions. To do this, Kepler found that he had to break with an ancient prejudice about planetary motions.

Elliptical Orbits and Kepler’s First Law

Astronomers had long assumed that heavenly objects move in circles, which were considered the most perfect and harmonious of all geometric shapes. They believed that if a perfect God resided in heaven along with the stars and planets, then the motions of these objects must be perfect too. Against this context, Kepler dared to try to explain planetary motions with noncircular curves. In particular, he found that he had the best success with a particular kind of curve called an ellipse.

You can draw an ellipse by using a loop of string, two thumbtacks, and a pencil, as shown in Figure 4-10a. Each thumbtack in the figure is at a focus (plural foci) of the ellipse; an ellipse has two foci. The longest diameter of an ellipse, called the major axis, passes through both foci. Half of that distance is called the semimajor axis and is usually designated by the letter a. A circle is a special case of an ellipse in which the two foci are at the same point (this corresponds to using only a single thumbtack in Figure 4-10b). The semimajor axis of a circle is equal to its radius.

Figure 4-10: Ellipses (a) To draw an ellipse, use two thumbtacks to secure the ends of a piece of string, then use a pencil to pull the string taut. If you move the pencil while keeping the string taut, the pencil traces out an ellipse. The thumbtacks are located at the two foci of the ellipse. The major axis is the greatest distance across the ellipse; the semimajor axis is half of this distance. (b) A series of ellipses with the same major axis but different eccentricities. An ellipse can have any eccentricity from e = 0 (a circle) to just under e = 1 (virtually a straight line).

By assuming that planetary orbits were ellipses, Kepler found, to his delight, that he could make his theoretical calculations match precisely to Tycho’s observations. This important discovery, first published in 1609, is now called Kepler’s first law:

The orbit of a planet about the Sun is an ellipse with the Sun at one focus.

The semimajor axis a of a planet’s orbit is the average distance between the planet and the Sun.

Ellipses come in different shapes, depending on the elongation of the ellipse. The shape of an ellipse is described by its eccentricity, designated by the letter e. Figure 4-10b shows a few examples of ellipses with different eccentricities. The value of e can range from 0 (a circle) to just under 1 (nearly a straight line). The greater the eccentricity, the more elongated the ellipse. Because a circle is a special case of an ellipse, it is possible to have a perfectly circular orbit. But all of the objects that orbit the Sun have orbits that are at least slightly elliptical. The most circular of any planetary orbit is that of Venus, with an eccentricity of just 0.007; Mercury’s orbit has an eccentricity of 0.206, and a number of small bodies called comets move in very elongated orbits with eccentricities of less than 1.

Orbital Speeds and Kepler’s Second Law

Once he knew the shape of a planet’s orbit, Kepler was ready to describe exactly how it moves on that orbit. As a planet travels in an elliptical orbit, its distance from the Sun varies. Kepler realized that the speed of a planet also varies along its orbit. A planet moves most rapidly when it is nearest the Sun, at a point on its orbit called perihelion. Conversely, a planet moves most slowly when it is farthest from the Sun, at a point called aphelion (Figure 4-11). After much trial and error, Kepler found a way to describe just how a planet’s speed varies as it moves along its orbit. Figure 4-11 illustrates this discovery, referred to as Kepler’s second law:

A line joining a planet and the Sun sweeps out equal areas in equal intervals of time.

Figure 4-11: Kepler’s First and Second Laws According to Kepler’s first law, a planet travels around the Sun along an elliptical orbit with the Sun at one focus. According to his second law, a planet moves fastest when closest to the Sun (at perihelion) and slowest when farthest from the Sun (at aphelion). As the planet moves, an imaginary line joining the planet and the Sun sweeps out equal areas in equal intervals of time (from A to B or from C to D). By using these laws in his calculations, Kepler found an excellent fit to the apparent motions of the planets.

This relationship is also called the law of equal areas. In the idealized case of a circular orbit, a planet would have to move at a constant speed around the orbit in order to satisfy Kepler’s second law. In the case of an elliptical orbit, a planet speeds up when it is closer to the Sun, and slows down when it is farther away.

Orbital Periods and Kepler’s Third Law

Kepler’s second law describes how the speed of a given planet changes as it orbits the Sun. Kepler also deduced from Tycho’s data a relationship that can be used to compare the motions of different planets. Published in 1618 and now called Kepler’s third law, it states a relationship between the size of a planet’s orbit and the time the planet takes to go once around the Sun:

The square of the sidereal period of a planet is directly proportional to the cube of the semimajor axis of the orbit.

Kepler’s third law says that the larger a planet’s orbit—that is, the larger the semimajor axis, or average distance from the planet to the Sun—the longer the sidereal period, which is the time it takes the planet to complete an orbit. From Kepler’s third law one can show that the larger the semimajor axis, the slower the average speed at which the planet moves around its orbit. (By contrast, Kepler’s second law describes how the speed of a given planet is sometimes faster and sometimes slower than its average speed.) This qualitative relationship between orbital size and orbital speed is just what Aristarchus and Copernicus used to explain retrograde motion, as we saw in Section 4-2. Kepler’s great contribution was to make this relationship a quantitative one.

It is useful to restate Kepler’s third law as an equation. If a planet’s sidereal period P is measured in years and the length of its semimajor axis a is measured in astronomical units (AU), where 1 AU is the average distance from Earth to the Sun (see Section 1-7), then Kepler’s third law is

Kepler’s third law

• P = planet's sidereal period, in years
• a = planet's semimajor axis, in AU

If you know either the sidereal period of a planet or the semimajor axis of its orbit, you can find the other quantity using this equation. Box 4-2 gives some examples of how this is done.

We can verify Kepler’s third law for all of the planets, including those that were discovered after Kepler’s death, using data from Table 4-1 and Table 4-2. If Kepler’s third law is correct, for each planet the numerical values of P² and a³ should be equal. This result is indeed true to very high accuracy, as Table 4-3 shows.

The Significance of Kepler’s Laws

Kepler’s laws are a landmark in the history of astronomy. They made it possible to calculate the motions of the planets with better accuracy than any geocentric model ever had, and they helped to justify the idea of a heliocentric model. Kepler’s laws also pass the test of Occam’s razor, for they are simpler in every way than the schemes of Ptolemy or Copernicus, both of which used a complicated combination of circles.

But the significance of Kepler’s laws goes beyond understanding planetary orbits. These same laws are also obeyed by spacecraft orbiting Earth, by two stars revolving about each other in a binary star system, and even by galaxies in their orbits about each other. Throughout this book, we shall use Kepler’s laws in a wide range of situations.

As impressive as Kepler’s accomplishments were, he did not prove that the planets orbit the Sun, nor was he able to explain why planets move in accordance with his three laws. These advances were made by two other figures who loom large in the history of astronomy: Galileo Galilei and Isaac Newton.